OK, that makes more sense - we can now guess further 🙂
Seems like the signal sense circuitry is working, as they stay in standby in Powerlink mode with nothing connected, and turn on with a cable connected (due to stray interference I suppose).
It could be that the signal path from the RCA connector is broken either at the connector or the selector switch. It is odd that both speakers would be like that... It is also strange that you don't hear any kind of hum or buzz with the cable end open, though. I believe we are ready to ditch the seller's claim that they worked 🙁
The thing is these speakers have two power supplies. The standby power supply, which is on all the time, keeps the signal sense circuitry powered which also controls the indicator LED. When it sees incoming audio (or the Powerlink power on signal) it pulls a relay, which turns on the main power supply to the amplifiers. You should hear the relay click when the speaker turns on & off; unfortunately it isn't too loud and is buried inside the metal casing, so it isn't quite easy.
After that, there is still a thermal fuse for the main supply transformer and two fuses on its secondary... any of these could be blown, preventing the amp from getting power. But once again I wonder how it could be the problem on both of the speakers.
BTW the switch is up L, middle R, down RCA.